As pp have said, work out first what your visa position would be. It sounds as if your DH would be able to apply for an intra-company transfer L1 visa, but they have to demonstrate that they need to bring him over rather than employ someone already in the US, and he needs to be working at the company for a number of years and be at a senior level. Even if it's all straightfoward it can still take some time to get the paperwork together and complete the visa application process (you have to go in person to the embassy in London too)
Your questions:
How much would we need monthly to live a comfortable lifestyle?
About $6000 after tax would get you a nice place to live, a decent car and the ability to travel around the US a bit too
Would we need to consider private school or are the state schools good?
Massachusetts has some of the best schools in the US, and some of the towns like Brookline and Newton have some of the best schools in MA. There is a helpful website called greatschools.org which has useful information about individual schools, but remember that there are still significant differences between the US and UK school systems. Depending on the age of your DC, you might prefer being on particular school districts, for example some are Kindergarten to Grade 6, then middle school then high school; some are K to Grade 8 then high school.
Yes, your school is determined by your address in most places - there is a lottery allocation in Boston plus selective admission. Personally I would avoid this like the plague, as the good schools are great but the bad schools are awful.
- Where is the best place to look for apartments?
Do you mean websites? Try Zillow and Trulia for listings, but you will probably want to find a realtor to work on your behalf to help find you an apartment and complete the paperwork. Ideally your DH's company would pay for this service as part of the overall relocation package.
If you mean locations, I would think about which school district you want to be in and then work from there. Depending on where in Cambridge DH needs to be, you could consider living in Brookline and he can easily cycle to work (many people do this), or out in Arlington and drive in (ditto). Cambridge itself is nice, very vibrant and lots of great places to eat out, but schools can be hit and miss I believe.
Any other things to consider?
Health insurance. The single most important thing (once your visas are arranged) to have in place and understand. It could cost you quite a bit each month, so be prepared for that when doing your sums, but it's essential.
There was a thread on here quite recently about an OP considering a move to New York - there were lots of thoughts on there too about the things that need consideration
Am I crazy for even thinking about it?
No!! It's a fabulous part of the world to live, especially for families. In an hour's drive you can be skiing in the winter and on the most beautiful beaches on the Cape in the summer, but Boston itself (and metro Boston, as the surrounding area is known) is so great you might not want to leave. 6 hour flight to get home and some decent ticket prices too (Norwegian and IcelandAir fly from Boston, as well as Virgin, Aer Lingus, BA, American etc etc etc)